Northeastern (46th Annual) and North-Central (45th Annual) Joint Meeting (20–22 March 2011)

Paper No. 1
Presentation Time: 8:00 AM-12:00 PM

BEST-FIT STRAIN FROM MULTIPLE ANGLES OF SHEAR AND IMPLEMENTATION IN A COMPUTER PROGRAM FOR GEOLOGICAL STRAIN ANALYSIS


VOLLMER, Frederick W., Geology, SUNY New Paltz, New Paltz, NY 12401, vollmerf@newpaltz.edu

An analytic solution to the Wellman (1962) construction was desired for a 2D and 3D strain analysis program, EllipseFit 2, designed for field-based strain studies and structural geology labs. Oriented images are digitized using center-points, five-point ellipses, polygon-moment ellipses, or line segment pairs. Techniques include center-to-center (Fry, 1979; Erslev, 1988) with ellipse-fitting, Rf/Φ (Dunnet, 1969) and polar (Elliott, 1970) graphs. Best-fit ellipse calculations include shape-matrix eigenvalues (Shimamoto and Ikeda, 1976), mean radial length (Mulchrone et al., 2003), and hyperboloidal vector mean (Yamaji, 2008), with error analysis. Fitting of section-ellipses to ellipsoids is done using Shan's (2008) method.

A Wellman construction (1962) is used to determine strain from pairs of initially orthogonal lines. Ragan and Groshong (1993) gave a trigonometric solution for two angles of shear, however the tangent function makes this numerically unstable, so an analytic geometry solution was derived. For two line segments, p with endpoints (x0, y0)T and (x1, y1)T, and q with endpoints (x2, y2)T and (x3, y3)T, p and q are translated: p' = p - (x0 + 1, y0)T, q' = q - (x2 - 1 , y2)T so one endpoint of p lies at (-1, 0)T, and one endpoint of q lies at (1, 0)T. The implicit forms for the two translated lines are: a1x + a2y + a3 = 0, b1x + b2y + b3 = 0, where: a = (y0' – y1', x1' – x0', x0'y1' – x1'y0')T, b = (y2' – y3', x3' – x2', x2'y3' – x3'y2')T. Solving these gives two diametrically opposed points x, y and -x, -y on the ellipse: x = (b1c2 – b2c1) / (a1b2 – a2b1), y = (a2c1 – a1c2) / (a1b2 – a2b1). For two pairs of line segments the strain can be solved directly. For two or more pairs the best-fit ellipse can be found by a least-squares minimization directly or by a LU decomposition to solve for the coefficients of the quadratic equation of a centered ellipse.